FRANKFURT—Volkswagen AG said on Monday it would offer a $1,000 goodwill package to U.S. customers whose diesel vehicles are affected by the emissions scandal as a “first step” in the German car maker’s effort to rebuild trust.

The announcement confirms a report in The Wall Street Journal about the company’s plans to boost customer loyalty nearly two months after revelations that Europe’s biggest car maker by sales engaged in widespread deception on emissions tests in the U.S. and Europe.